Trigonometric Fourier series

The trigonometric Fourier series is a mathematical representation of a periodic time function using cosine and sine terms. It allows us to express any periodic function as a sum of sinusoidal functions with different frequencies and amplitudes. The general form of a trigonometric Fourier series is given by:

f(t) = a0 + Σ(an * cos(nωt) + bn * sin(nωt))

where f(t) is the periodic function, t is the time, a0 is the DC (direct current) term, an and bn are the Fourier coefficients, n is the harmonic number, and ω is the angular frequency.

DC term

The DC term represents the average value of the periodic function over one period. It is a constant term that is unaffected by the frequency components of the function. In the trigonometric Fourier series, the DC term is represented by a0.

Cosine terms

The cosine terms in the series represent the even components of the periodic function. They have frequencies that are integer multiples of the fundamental frequency, ω. The coefficients an determine the amplitudes of the cosine terms.

Sine terms

The sine terms in the series represent the odd components of the periodic function. They also have frequencies that are integer multiples of the fundamental frequency, ω. The coefficients bn determine the amplitudes of the sine terms.

Explanation of the answer

The correct answer is option 'C', which states that the trigonometric Fourier series of a periodic time function can have only cosine terms. This means that the function being represented does not have any odd components or sine terms in its Fourier series.

The absence of sine terms implies that the function is symmetric about the vertical axis or has an even symmetry. In such cases, the Fourier coefficients bn representing the amplitudes of the sine terms will be zero.

Therefore, the trigonometric Fourier series will only contain cosine terms, which represent the even components of the function. This is consistent with the even symmetry of the function.

It is important to note that the absence of sine terms does not mean that there are no odd components in the function. It simply means that these odd components can be expressed as combinations of the cosine terms using trigonometric identities.

In summary, the trigonometric Fourier series of a periodic time function can have only cosine terms when the function has an even symmetry or is symmetric about the vertical axis.